Problem: Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1}.\]
Solution: If $\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1},$ then
\[\begin{pmatrix} a & b \\ c & d \end{pmatrix} \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \frac{1}{a} & \frac{1}{b} \\ \frac{1}{c} & \frac{1}{d} \end{pmatrix} \renewcommand{\arraystretch}{1} = \mathbf{I}.\]This becomes
\[\renewcommand{\arraystretch}{1.5} \begin{pmatrix} 1 + \frac{b}{c} & \frac{a}{b} + \frac{b}{d} \\ \frac{c}{a} + \frac{d}{c} & \frac{c}{b} + 1 \end{pmatrix} \renewcommand{\arraystretch}{1} = \mathbf{I}.\]Then $1 + \frac{b}{c} = 1,$ so $\frac{b}{c} = 0,$ which means $b = 0.$  But then $\frac{1}{b}$ is undefined, so there are $\boxed{0}$ solutions.